A family of relative difference sets associated with Hjelmslev structures over finite projective spaces

In this note, we construct a new family of relative difference sets, with parameters n=q^d, m=q^d+...+q+1, k=q^d-1(q^d-1), ₁ =q^d-1(q^d-q^d-1-1) and ₂ =q^d-2(q-1)(q^d-1-1) where q is a prime power and d 2 an integer. The associated symmetric divisible designs admit natural epimorphisms onto the symmetric designs formed by points and hyperplanes in the corresponding projective spaces PG(d,q). As in the theory of Hjelmslev planes, points with the same image can be recognized from having the larger of the two possible joining numbers, and dually. More formally, these symmetric divisible designs are balanced c-H-structures (in the sense of Drake and Jungnickel 2]) with parameters c=q^d-2 (q-1)² and t=q^d-1 (q-1) over PG_d-1(d,q). These are the first examples of balanced non-uniform c-H-structures of type 2; they can be used in known constructions to obtain new balanced c-H-structures (for suitable c) of arbitrary type. In fact, all these results are special cases of a more general construction involving arbitrary difference sets.The author gratefully acknowledges the hospitality of the University of Waterloo and the financial support of NSERC under grant IS-0367.